Clearly, the constraint \(n\cdot n=v^2\)  makes the \(O(4)\)  symmetry to be realized in a non-linear way.
This comes about because the four components of \(n=(\omega ^1,\omega ^2,\omega ^3,\pm \sqrt{v^2-{\omega }^2})\) 
rotate linearly with \(O\)  in Eq. (REF ),
imposing a non-linear transformation law on the polar-coordinate Goldstone bosons, \(w^a\)  (\(a=1,2,3\) ). In these polar coordinates, the Higgs sector SM Lagrangian is Note the difference with Eq.(2.12) in [1]} where the \(\omega _i\omega _j\)  piece is absent. It is unnecessary unless amplitudes with more than two Goldstone bosons are analyzed, which we leave for future investigation.
\(\mathcal {L}_{SM}=\frac{1}{2}\Big (1+\frac{h}{v}\Big )^2(\partial _\mu \omega ^i\partial _\mu \omega ^j )\left(\delta _{ij}+\frac{\omega _i\omega _j}{v^2-\omega ^2}\right)+\frac{1}{2}(\partial _\mu h)^2-\frac{\lambda }{4} (h^2+2vh)^2\;.\) 
